Starting from the merger rate, we follow the standard procedure to derive the SGWB formed by overlapped GWs of compact binaries, see e.g. [1]}. The present day energy fraction of the SGWB is defined by \(\Omega _\text{\tiny GW} \equiv (d \rho _{\rm GW}/ d\ln f)/\rho _c\) , where \(\rho _{\rm GW}\)  is the energy density of GWs, and it is given by the following integral:
\(\Omega _\text{\tiny GW} (f)= \frac{f}{\rho _c} \int _0^{z_{\rm max}} \mathrm {d}z^{\prime }\, \mathrm {d}m_1 \,\mathrm {d}m_2 \,\frac{1}{(1+z^{\prime })H(z^{\prime })}\times \frac{\mathrm {d}^2 \tau _{\rm merg}(z^{\prime },\,m_1,\,m_2)}{\mathrm {d}m_1 \mathrm {d}m_2} \frac{\mathrm {d}E_\text{\tiny GW} (f_s)}{\mathrm {d}f_s},\) 
